Let Y be a topological space
and Z be a metric space with metric d(⋅,⋅). Denote by C(Y,Z) the space of all
continuous functions from Y into Z. For a given topological space X and a point wise
continuous mapping T : X → C(Y,Z) a theorem is proved asserting (under some
conditions) that T is continuous at the points of some dense Gδ subset of X with
respect to the topology of uniform convergence in C(Y,Z). A “set-valued” version of
this result is also proved. It is shown how one can use these results in order to
get new information about points of continuity and single-valuedness of
(multivalued) monotone operators and (multivalued) metric projections. As
corollaries some known results about Gâteaux or Fréchet differentiability
of convex functions on a dense subset of their domains of continuity are
obtained.
